The extended semidefinite linear complementarity problem: A reformulation approach (Q2781089)

It is considered the problem: find \((x,y)\in S^m \times S^m\) such that \(Mx-Ny\in C\) \(x\in K^m\), \(y\in K^m\langle x,y\rangle =0\) (XSDLCP) where \(K^m\) denotes the closed convex cone of \(m\times m\) positive semidefinite symmetric matrices, \(S^m\) the linear space of the \(m\times m\) real symmetric matrices and \(\langle\cdot, \cdot\rangle\) the inner product defined by the trace of \(A.B\). \(C=\{u\in S^n\mid Au-b \in K^k\}\). \(M,N:S^m\to S^n\) are linear mappings, \(A:S^n\to S^k\) linear mapping, \(b\in S^k\). It is assumed that the feasible set of the problem is nonempty. The authors propose 2 merit functions which allows the reformulation of (XSDLCP) as optimization problem. The functions are the sum of a smooth penalization of \(C\) and either the squared Fisher-Burmeister or the implicit Lagrangian function. Conditions are given under which any stationary point of the ment function solves (XSDLCP).NEWLINENEWLINEFor the entire collection see [Zbl 0969.00060].